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Inhomogeneous percolation problems and incipient infinite clusters

Publication ,  Journal Article
Chayes, JT; Chayes, L; Durrett, R
Published in: Journal of Physics A: Mathematical and General
December 1, 1987

The authors consider inhomogeneous percolation models with density p c+f(x) and examine the forms of f(x) which produce incipient structures. Taking f(x) approximately= mod x mod - lambda and assuming the existence of a correlation length exponent v for the homogeneous percolation model, they prove that in d=2, the borderline value of lambda is lambda b=1/v. If lambda >1/v then, with probability one, there is no infinite cluster, while if lambda <1/v then, with positive probability, the origin is part of an infinite cluster. This result sheds some light on numerical and theoretical predictions of certain properties of incipient infinite clusters. Furthermore, for d>2, the models studied suggest what sort of 'incipient objects' should be examined in random surface models.

Duke Scholars

Published In

Journal of Physics A: Mathematical and General

DOI

ISSN

0305-4470

Publication Date

December 1, 1987

Volume

20

Issue

6

Start / End Page

1521 / 1530

Related Subject Headings

  • Mathematical Physics
  • 02 Physical Sciences
  • 01 Mathematical Sciences
 

Citation

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Chayes, J. T., Chayes, L., & Durrett, R. (1987). Inhomogeneous percolation problems and incipient infinite clusters. Journal of Physics A: Mathematical and General, 20(6), 1521–1530. https://doi.org/10.1088/0305-4470/20/6/034
Chayes, J. T., L. Chayes, and R. Durrett. “Inhomogeneous percolation problems and incipient infinite clusters.” Journal of Physics A: Mathematical and General 20, no. 6 (December 1, 1987): 1521–30. https://doi.org/10.1088/0305-4470/20/6/034.
Chayes JT, Chayes L, Durrett R. Inhomogeneous percolation problems and incipient infinite clusters. Journal of Physics A: Mathematical and General. 1987 Dec 1;20(6):1521–30.
Chayes, J. T., et al. “Inhomogeneous percolation problems and incipient infinite clusters.” Journal of Physics A: Mathematical and General, vol. 20, no. 6, Dec. 1987, pp. 1521–30. Scopus, doi:10.1088/0305-4470/20/6/034.
Chayes JT, Chayes L, Durrett R. Inhomogeneous percolation problems and incipient infinite clusters. Journal of Physics A: Mathematical and General. 1987 Dec 1;20(6):1521–1530.

Published In

Journal of Physics A: Mathematical and General

DOI

ISSN

0305-4470

Publication Date

December 1, 1987

Volume

20

Issue

6

Start / End Page

1521 / 1530

Related Subject Headings

  • Mathematical Physics
  • 02 Physical Sciences
  • 01 Mathematical Sciences